Group #34100957 — subgroups

Subgroups — #34100957

Order 20096.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097695	C₂	1	4	yes	maximal
2	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
3	#23757139	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
7	#19097695	—	2	8	no	—
8	#19097682	—	2	2	no	—
9	#19097678	—	2	2	no	—
10	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
12	C₄⋊Q₈×C₁₅₇	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₅₇	—	2	2	no	—
14	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₅₇	—	2	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	—	2	2	no	—
18	C₂×#11929434	D₄	1	4	yes	—
19	C₄⋊Dic₁₅₇	—	2	4	no	—
20	C₂×Q₈×C₁₅₇	—	2	2	no	—
21	C₂×Q₈×C₁₅₇	—	2	4	no	—
22	C₂×Q₈×C₁₅₇	—	2	4	no	—
23	C₂×Q₈×C₁₅₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₅₇	—	2	2	no	—
25	C₂²×C₄×C₁₅₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₅₇	—	2	8	no	—
28	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₅₇	D₄	1	2	yes	—
30	C₂×Dic₁₅₇	—	4	2	no	—
31	C₂×Dic₁₅₇	—	4	4	no	—
32	#11929434	—	2	4	no	—
33	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₅₇	—	2	8	no	—
35	Q₈×C₁₅₇	—	4	4	no	—
36	C₂×C₄×C₁₅₇	—	2	4	no	—
37	C₂×C₄×C₁₅₇	—	2	4	no	—
38	C₂×C₄×C₁₅₇	—	4	2	no	—
39	C₂×C₄×C₁₅₇	—	4	2	no	—
40	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₇	#41	1	2	yes	—
45	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₅₇	#124	1	1	yes	characteristic
47	C₂²×C₁₅₇	#127	1	1	yes	characteristic
48	C₂²×C₁₅₇	#137	1	4	yes	—
49	C₄×C₁₅₇	—	2	4	no	—
50	C₄×C₁₅₇	—	4	4	no	—
51	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₅₇	#121	1	2	yes	—
53	Dic₁₅₇	—	8	2	no	—
54	C₂×C₁₅₇	#390	1	1	yes	characteristic
55	C₂×C₁₅₇	#419	1	2	yes	—
56	C₂×C₁₅₇	#491	1	2	yes	—
57	C₂×C₁₅₇	#497	1	2	yes	—
58	C₁₅₇	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	157	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
61	C₂×#103	—	157	4	no	—
62	C₂×#97	—	157	1	no	—
63	#382	—	157	1	no	—
64	C₂²×Q₈	C₂×D₁₅₇	1	2	yes	—
65	C₂²×C₈	—	157	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₅₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	157	2	no	—
70	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
71	#103	—	314	8	no	—
72	#97	—	314	2	no	—
73	#95	—	314	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₅₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	314	2	no	—
80	C₂²×C₄	C₂²⋊D₁₅₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
82	C₂×C₈	—	314	2	no	—
83	C₂×C₈	—	314	2	no	—
84	C₂×C₈	—	157	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	314	4	no	—
87	C₄²	C₂×Dic₁₅₇	1	2	yes	—
88	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	628	2	no	—
97	C₂×C₄	—	628	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₅₇	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₅₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
102	C₂×C₄	#14578953	1	2	yes	—
103	C₈	—	314	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
105	C₂²	#19097758	1	1	yes	characteristic
106	C₂²	#19097767	1	1	yes	characteristic
107	C₂²	#19097802	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1256	2	no	—
111	C₄	D₄×Dic₁₅₇	1	2	yes	—
112	C₄	#19097742	1	2	yes	—
113	C₂	#23757176	1	1	yes	characteristic
114	C₂	#23757282	1	2	yes	—
115	C₂	#23757641	1	2	yes	—
116	C₂	#23757665	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097695

C₂

yes

maximal

C₂×#19097682

C₂

yes

maximal, characteristic

#23757139

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097695

—

#19097682

—

#19097678

—

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

D₄

yes

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂×C₄

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

#121

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#390

yes

characteristic

C₂×C₁₅₇

#419

yes

—

C₂×C₁₅₇

#491

yes

—

C₂×C₁₅₇

#497

yes

—

C₁₅₇

#1960

yes

characteristic, complemented, Sylow

#1960

—

157

maximal, Sylow

C₂×C₄⋊Q₈

D₁₅₇

yes

characteristic

C₂×#103

—

157

—

C₂×#97

—

157

—

#382

—

157

—

C₂²×Q₈

C₂×D₁₅₇

yes

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₅₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#103

—

314

—

#97

—

314

—

#95

—

314

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₅₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄²

C₂×Dic₁₅₇

yes

—

C₄²

C₂²⋊D₁₅₇

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

628

—

C₂×C₄

—

628

—

C₂×C₄

C₂²⋊Dic₁₅₇

yes

—

C₂×C₄

C₂²×Dic₁₅₇

yes

characteristic

100

C₂×C₄

D₄×D₁₅₇

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

102

C₂×C₄

#14578953

yes

—

103

C₈

—

314

—

104

C₂²

C₂×C₂²⋊Dic₁₅₇

yes

characteristic

105

C₂²

#19097758

yes

characteristic

106

C₂²

#19097767

yes

characteristic

107

C₂²

#19097802

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1256

—

111

C₄

D₄×Dic₁₅₇

yes

—

112

C₄

#19097742

yes

—

113

C₂

#23757176

yes

characteristic

114

C₂

#23757282

yes

—

115

C₂

#23757641

yes

—

116

C₂

#23757665

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#1960
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

C₂×C₄⋊Q₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₅₇

hypocenter

C₁₅₇

#1960

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₅₇	C₄	semidirect
58. C₁₅₇	#1960	semidirect
67. C₄⋊Q₈	Dic₁₅₇	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₅₇

C₄

semidirect

58. C₁₅₇

#1960

semidirect

67. C₄⋊Q₈

Dic₁₅₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19097695	2
C₂×#19097682	2
#23757139	2
C₂×C₄⋊Q₈×C₁₅₇	2
#1960	157

subgroup

index

C₂×C₄⋊Q₈×C₁₅₇

157

Subgroups — #34100957

Conjugacy classes of subgroups

Complements

Maximal subgroups